Among a number of small and often strange ­humanoid sculptures ­sitting on low plinths in Tate Britain's Henry Moore exhibition, there is an object that made me laugh out loud. But it was a laugh tempered by ­uncertainty. A small slabby thing carved from warm Corsehill stone, the sculpture appears to represent a man lying in a bath, his arms at his sides, staring down at his own erection.

It's the sort of universal situation any man might recognise. The head, at one end of the slab, is like a grave-stone and pierced by a single hole. At the other end is a rounded bulge, from which protrudes an elongated wedge-like ridge, riding up his torso and terminating between two bumpy little breasts. Maybe it's not a man after all, nor a person in a bath, but some sort of woman. This is yet another of Moore's reclining figures, from 1933-4, but it could be a funerary slab from some ­ancient, even alien civilisation. What on earth was Moore thinking?

Another sculpture nearby, carved in the same stone, is most definitely female, this time upright – a rounded, almost jug-like form with two pert breasts. Woman as vessel, then. Yet the head, again pierced by a single hole, is almost penile. So, too, is the head of a third sculpture, a head-and-shoulders carved in African wonderstone, a glassy black volcanic rock, part of which is striated with faint concentric bands of colour, spreading like little seismic ripples. The entire sculpture seems to grow from these rings. It's a weighty thing that you want to touch and hold, to feel its mass, its sexy, bulging, androgynous smoothness.

These smaller sculptures, which Moore produced throughout his long career, repay all the attention you give them. The longer you look, the stranger they are. There are dismembered bodies, bowling-ball heads that want to eat smaller heads, holes that want to accept protruberences, body parts that morph into other body parts. One translucent alabaster sculpture describes a baby's head suckling at a breast – or does the breast suckle at the child? The whole thing has a sort of yearning, merging feel, and reminds me of the sculpted heads of Medardo Rosso, who died in 1928, two years ­before Moore carved this.

A lot of what Moore did has an almost feminine feel. This is not just because he made so many sculptures of mothers and infants, nor because (as he admitted) he had a bit of a mother fixation. It is as much in his play of insides and outsides, in the flow of forms and space. There is also something inescapably phallic about many of Moore's women, and in the male and female dualities even of his ­abstract forms. Moore never read Freud, though throughout the 1920s and 30s the discoveries of psychoanalysis were very much in the air.

Moore's smaller sculptures can be threatening, too. A number of little reclining figures from the end of the 1930s are made from lead, a material that attracted Moore because of its ­poisonous nature. The small bodies look as if they have been licked into shape. Light flows queasily over their smooth grey surfaces. It is as if they have been infected by Salvador Dalí and the bad android in Terminator 2. But what Moore appears to have been infected by, mostly, was Picasso – whose biographer, John Richardson, said that Moore was the "petit-maitre of Picasso's bone-surreal", a cruel but fitting jibe. Giacometti's surrealist work, as well as that of Jean Arp and other European artists have left their mark. Moore looked long and hard at a great deal of what inspired other artists of the period: African carvings; Mayan, Egyptian and early Iberian sculpture – art that was called "primitive". He ­buried himself in the British Museum and the Trocadero in Paris, knew ­Giacometti, and visited Picasso when the latter was painting Guernica.

Morbid and sexual undertones

So used are we to Henry Moore, we hardly give him the time his art ­deserves. This aim of this exhibition, which takes us from the 1920s up to the 1970s, is to show us his morbid and ­sexual undertones. The show avoids much of Moore's later work, when the artist went into production mode, making a form of sculpture that blinds us to his real achievements. All those public things on plinths, from Harlow New Town to Tokyo; all those British Council-sponsored exhibitions that ­forever circumnavigate the globe. Moore's bronze editions of ­interlocking, rounded forms with their hollows and holes and fussy, scaly surfaces have ­become so much street furniture.

The impression I grew up with was of a domesticated modernist, already old hat, so ubiquitous one almost didn't need to look. He was just there. It is the artist's smaller, more private works I like the best, the ones that don't appease airy humanist ­sentiment. The exhibition's curator, Chris Stephens, also hopes to present a Moore whose preoccupation was the human body as, in Stephens's words, "abject, erotic, vulnerable, violated and visceral . . . absurd, uncanny and claustrophobic". Ooh-err, what's come over our Henry, king of the sucked-toffee blob in the town square? The mother-and-child-friendly sculptor?

What Stephens really wants to do is give us a Moore for our time. A Moore who might hang out with Georges Bataille and the dissident ­surrealist crowd, a Moore to set ­beside Giacometti and Hans Bellmer, or even Louise Bourgeois, Paul ­McCarthy, Bruce Nauman and Thomas Schütte. It's true that Nauman has nodded to Moore in a couple of works, while Schütte's reclining steel women owe some of their deformations to Moore. What began as jokes at Moore's ­expense in the sausage ­sculptures of Fischli and Weiss, and in the ­performances of Bruce McLean, has ended as a kind of homage.

Sympathy for elms

There is a falling off in Moore's post-second world war work, although the reclining figure he made for the 1951 Festival of Britain remains a ­peculiar, startling and troubling thing. It is not a million miles from Francis ­Bacon's 1944 Studies at the Base of a ­Crucifixion – if one could imagine those figures spending a day at the beach. But it is in the last room – just when the show, and Moore's art, seems to flag – that things really come alive again. Four great elm carvings ­dominate the space, filling the air with their lovely reflected light. Each, again, is a reclining figure. The first is dated 1936, the last 1976-8. In this room, their scale seems just right. There is a sense of massive slumber and waiting.

But what really strikes me is Moore's craftsmanship, his understanding and sympathy towards the great hunks of elm, the way he followed the grain and density of the wood when he hollowed space and revealed form. He let the material do the talking, and respected its nature. People don't talk much of "truth to materials" nowadays. I don't really go in for all the mythic ­qualities these recumbent figures might have, but they do have real presence. All the smoothing and rounding and ­hollowing of the forms, even their facelessness, has a point. There is a sense of great gravity and rest. The sculptures slow time down to a full stop, and us with it.